Two workers can complete a job in 24 days while working together.

Question

Two workers can complete a job in 24 days while working together. First one of the two workers works alone for 16 days then the other worker works for 24 days alone. If it is known that only 20% of assigned work is left after 40 days, then find out the time (in days) taken by the slower worker to complete the remaining work.

A. 12
B. 14
C. 16
D. 17
E. 18

A. 12
B. 14
C. 16
D. 17
E. 18

imeanup · Accepted Answer

Method 1: Efficiency

A worked for  16  days (individually) and B worked for  24  days (individually) to complete  80\%  of the total work.

Let's find the efficiency of A, B. LCM (Lowest common factor) of  lcm(A, B) = lcm(16, 24) = 48

Efficiency of  A = \dfrac{48}{16} = 3   and  B = \dfrac{48}{24} = 2 . Note: that efficiency remains same even if they complete any percenatge of work.

Now find the total work, i.e.  	ext{Total work} = 	ext{Total Efficiency} 	imes 	ext{Time to complete the work together} = (3+2) 	imes 24 = 120  is the total work of which  80\%  was completed by A and B, i.e.  120 	imes \dfrac{80}{100} = 96 . Thus,  120-96 = 24  part of the work is remaining which was completed by B.

Time taken by B to complete the  24  part of the work, i.e.  \dfrac{	ext{Remaining work}}{	ext{B's efficiency}} = \dfrac{24}{2} = 12

A

PS: If you don't understand, REPEAT!

MuszerN · Answer

Let the rate of the first worker be 'a' and of the second worker be 'b'. The two workers working consecutively for the given durations complete 80% of the job. This gives us the following equation:
16a + 24b = 80/100
2a + 3b = 1/10

Furthermore, the two workers working together for 40 days at a rate of 1/24 complete 10/6 jobs. This allows us to set up the following equation:
40a + 40b = 10/6
4a + 4b = 1/6

Both equations together allow us to solve for the individual rates of the workers:
a = 1/40
b = 1/60

Let 'd' be the number of days it takes the slower worker to complete the remaining job. This gives us:
b*d = 1/5
d = 12

btsaami · Answer

Let A be eff. of 1st worker and B be eff. of the other worker.
From the data, 
80% of 24*(A+B)= 16*A + 24*B
4/5* 6 (A+B) = 4A+6B
4A=6B
A/B= 3/2 Hence B is the slower one.

1/5*24*(5)=2* D days
D= 12 days.

Hi,  Bunuel   KarishmaB  - Can you please clarify below point?

- On first reading the ques, I first assumed that the slower worker will finish the remaining task as it mentions " find out the time (in days) taken by the slower worker to complete the remaining work" but this kind of assumption is wrong, right? It might be the case that faster one completes the remaining task.

-From the given, 16 and 24 days, can i directly get A/B= 3/2, i am not sure here as only some proportion of the work has been completed. what if that A completes less work in 16 days and B completes more work in 24 days?

These kind of questions are making me doubt my conceptual knowledge.

KarishmaB · Answer

Two workers together complete the job in 24 days. One worker worked for 24 days alone.
Another worker worked for only 16 days alone. Had he worked for 24 days too, the work would have got completed. But 20% of the work is remaining because he did not work for 8 days. Say rate of work of this guy is r

8r = 1/5
r = 1/40

Slower guy's rate of work = Combined rate - r = 1/24 - 1/40 = 1/60

To complete 1/5th of the work, time taken by this slower guy alone will be (1/5) / (1/60) = 12 days

Answer (A)

I don't know the source of this question but it is certainly a good one. Conceptual, but simple calculation.

KarishmaB · Answer

We are given that the slower guy completes the rest of the work so we must identify who is slower. If the faster guy alone were doing it, he would simply take 8 days to complete it since both need to work for exactly 24 days to complete the work. Whether they do it simultaneously or one after another is irrelevant. Since all options are greater than 8, the second guy is slower. Alternatively, consider that if both had the same rate, and the first guy had worked for 8 fewer days, only 1/6th of the work would be remaining. But 1/5th of the work is remaining. It means the first guy was faster.

They do not do the same amount of work in 16 days and 24 days respectively. Hence ratio of 3/2 has no meaning here.

imeanup · Answer

btsaami

-From the given, 16 and 24 days, can i directly get A/B= 3/2, i am not sure here as only some proportion of the work has been completed. what if that A completes less work in 16 days and B completes more work in 24 days?

The statement by  KarishmaB  is not correct "They do not do the same amount of work in 16 days and 24 days respectively. Hence ratio of 3/2 has no meaning here."

Here is my counter:

You may consider "time and work" as "ratio and proportions" as well. Therefore, when you consider  A = 3  and  B = 2 , then you are talking about efficiency and how efficient the worker is. Higher the efficiency, efficient the worker is or lesser time they take to complete the work. Now, when you take  A = 3  (efficiency), the amount of work completed by A in  16  days  = 16 	imes 3 = 48 . Similarly,  B = 24 	imes 2 = 48 . Therefore, they complete the  same amount of work . Hence, the efficiency is the ratio which has more meaning.

Please don't discourage any method. There are multiple ways to solve a problem. And "efficiency method" is easy to tackle difficult problem.

If you have a better counter to present, I would appreciate your feedback as a mathematics enthusiast.

Chibvike · Answer

Both the faster and slower worker completes 80% of the work at their respective constant rates, so I assumed they each completed 40% of the work at their respective rates

If it takes the slower worker 24 days to complete 40% of the work it should take them 24/2= 12 days to complete 20% of the work remaining

Answer choice; A

Not sure if my approach is correct but I had to logically think through this after being stuck at some point in using algebra

KarishmaB · Answer

meanup

Request you to read the question put forward by the learner carefully:

From the given, 16 and 24 days, can i directly get A/B= 3/2, i am not sure here as only some proportion of the work has been completed. what if that A completes less work in 16 days and B completes more work in 24 days?

The question clearly asks whether given time taken is 2:3 implies that their rates are 3:2. This happens only when we know that both did the same work. That is what the learner worries about too and that is what I pointed out too. And no, I am not trying to discourage anyone from using any method. I myself like to use innovative approaches and that is what I encourage too.

That is all I will say on this topic.

Two workers can complete a job in 24 days while working together.

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